Most effecient way to migrate related tables from one Microsoft Access Web App to another?

Most effecient way to migrate related tables from one Microsoft Access Web
App to another?

I've got two Microsoft Access 2013 Web Apps, hosted in SharePoint 2013 as
part of Microsoft Office365. Both databases contain roughly the same
tables and structure (with some minor tweaks).
We have two databases because our particular SharePoint server encountered
a fluke error and irreparably corrupted the UI for the first Access Web
App. In the end it was more cost-effective to just build a second Access
Web App from scratch, rather then try to reverse the corruption in the
original.
So now I have the duplicate created and functional, but I need to migrate
data. The SQL backend for the original Web App is still fully functional,
and the UI within Microsoft Access is also fully functional (it's the
SharePoint side that got broken). So with Microsoft Access 2013 and SQL
Management Studio available as my tools, what is the most efficient way to
move data from one database to another? (Excel can connect and pull data,
but can't insert records in the new database as far as I can tell, making
it relatively useless. Let me know if I'm missing something here.)
Some of the problems I run into are, I don't have owner, DBA, or SysAdmin
access to SQL - just data read and data write. I can't run any SQL queries
that alter the table, such as renaming a column or SET IDENTITY_INSERT
etc. (Was tinkering with setting the identity column, as one way of
preserving existing relationships between records.) Additionally, Access
2013 doesn't provide the full ribbon for Web Apps, but instead only
provides extremely limited functionality (screenshot below). Additionally,
the query and macro buttons do not allow you to create queries or macros
directly via code (SQL, VBA, etc.), but rather they force you to only use
the query wizard GUI and macro GUI, and they do not allow you to connect
to another database to pull data.
With these constraints, what is the best way to migrate existing data from
the first database into the second?